u/Dense_Toe_6976

I worked on this problem a bit before in earlier years to extend the domain past |s| < 1 and recently came back to it since learning more about infinite sums. I believe I came across an answer. I tried looking online for other solutions like this one and I haven’t been able to. If this solution has a name or something tied to it, I’d really appreciate knowing what it is if anyone knows (or if anyone knows of other representations). From what I can tell, it seems to converge for any complex number s where |s| > 1. Here’s the equation:

-ln(ln(s)) / ln(2) + sum(k >= 1, ln(s)^k / (k! * (1-2^k)))